4/18/2011. Titus Beu University Babes-Bolyai Department of Theoretical and Computational Physics Cluj-Napoca, Romania

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1 1. Introduction Titus Beu University Babes-Bolyai Department of Theoretical and Computational Physics Cluj-Napoca, Romania Bibliography Computer experiments Ensemble averages and time averages Molecular dynamics Relationship of MD statistical mechanics Monte Carlo method Relationship of MC statistical mechanics Metropolis sampling algorithm Differences between MD and MC Free Internet resources for MD 1

2 Rapaport, D.C., The Art of Molecular Dynamics Simulation. Second Edition (Cambridge University Press, 2004). Leach, A. R., Molecular Modelling. Principles and applications, Second edition (Prentice Hall, 2001). Frenkel, D., Smit, B., Understanding Molecular Simulation. From Algorithms to Applications (Academic Press, 2002). Allen, M. P., Tildesley, D. J., Computer Simulation of Liquids(Clarendon Press, Oxford, 1991). Sadus, R.J., Molecular Simulation of Fluids. Applications to Physical Systems, Second Edition (Elsevier, 1999). Griebel, M., Knapek, S., Zumbusch, G., Numerical Simulation in Molecular Dynamics. Numerics, Algorithms, Parallelization, Applications (Springer, 2007). Ercolessi, F., A molecular dynamics primer, Physical sciences interplay between experiment and theory Experiment numerical results by measurements Theoretical treatment: elaboration of a model mathematical equations (typically) validationof the model ability to describe the behaviorof the real system. Most physical problems analytically solvable after being simplified Simplified model may significantly deviate from reality 2

3 Computer experiment (simulation): Based on a theoretical model Calculations prescribed by a program, based on an algorithm Distinctive features: microscopic hazard plays a key role (like in real experiments) use of stochastic models based on random variables Computer virtual laboratory, rather than deterministic calculator. Problem complexity / model realism scale with computer performance Molecular simulations(generic term): Monte Carlo method Molecular Dynamics Computational statistical mechanics "exact" statistical mechanics results Macroscopic properties ensemble averages Computed observable quantities attributes of real measurements Statistical fluctuations interpreted by statistical methods Reproducibility statistical character 3

4 Ensemble infinite collection of identical (independent) copies of the system, each in one of the microscopic states compatible with the macroscopic state Ensemble average(expectation value): 1 N N N N N N A = d d A,, ensemble 3N N! h p r p r ρ p r A(p N,r N ) observable of interest, ρ(p N,r N ) probability density functions of momenta and positions of all system components ( ) ( ) Canonical (NVT) ensemble prototype is a system at constant temperature Probability density of the ensemble: ρ N N 1 N N ( p, r ) = exp H (, ) / kbt Q p r H(p N,r N ) Hamiltonian, T temperature, k B Boltzmann s constant Partition function: 1 N N N N Q = d d exp H 3 (, ) / k N BT N! h p r p r Measurable physical quantities are defined as time averages: 1 τ time lim ( N N A = A ( t), ( t) ) dt τ τ p r 0 Ergodic hypothesis the ensemble average is equal to the time average A = A time ensemble 4

5 Newton's equations of motion solved to generate new configurations Sampling the whole phase space along a continuous path Time-dpendent properties can be investigated. Typical applications: Fundamental studies: molecular chaos, kinetic theory, diffusion, transport properties Phase transitions: phase coexistence, order parameters, critical phenomena Collective behavior: space/time correlation functions, vibrations, spectroscopy Complex fluids: glasses, molecular and ionic liquids, films and monolayers Polymers: relaxation and transport properties Solids: defects, fractures, structural transformations, mechanical properties Fluid dynamics: laminar flow, boundary layers, unstable flows etc. MD simulates the real dynamics of the system time averages of properties Atomic positions derived in sequence from Newton's equationsof motion, propagated with typical time steps of 1-10 femtoseconds MD is a deterministic method system state along a trajectory can be predicted at any future time Improved statistics by averaging over ensembles of trajectoriesstarted from random initial conditions Time average: M 1 A = A r t p t time M m = 1 M number of time steps (measurements) N N ( ( m ), ( m )) Typical simulations at constant total energy microcanonical ensemble Other ensembles can be simulated by applying thermostats (NVT), barostats(npt) etc. 5

6 Stochastic strategy for studying systems in thermal equilibrium Dynamics intrinsically absent has to be prescribed Sampling microstates in configuration space update algorithm with probabilities Improved accuracy sampling with higher probability the significant microstates Importance sampling Metropolis algorithm(the most famous) Typical applications: Dense gases and liquids Phase transitions in systems of interacting variables (spin) ex. Ising model Strongly correlated quantum mechanical systems ex. Hubbard model Lattice gauge theories fundamental interactions in particle physics Tests of universality and finite-size scaling Theories of early universe Samples only the configuration space. Ensemble average integrals replaced by sums over discrete microstates: A s A se Es s e E s. Generation of microstates: uniform probability waste of effort (most of the states contribute little), importance sampling(art of MC) generating microstates with a given probability function for the macrocanonicalensemble, e -ßEs : S total number of sampled microstates S A 1 S s 1 A s, 6

7 1. Establish initial microstate 2. Make random trial change in microstate (ex. choose a spin (particle) at random and flip (displace) it) 3. Compute energy changeof system due to trial change, E = E trial E old 4. If E 0energy decrease acceptnew microstate; go to step 6 5. If E > 0energy increase accept with probability w = e -ß E compute the quantity w = e -ß E generate uniformly a random number 0 r < 1 If r w accept new microstate, otherwise retain previous one 6. Calculate physical quantities of interest and sums for averages 7. Periodically compute averages over microstates 8. Repeat steps 2-7 to obtain a sufficient number of microstates MD provides information about time dependence of properties MC no temporal relationship between successive Monte Carlo configurations MD it is possible to predict the configurationat any time in the future or past MC outcome of each trial move depends only upon its immediate predecessor MD kinetic and potential energy contribution to the total energy MC total energy is determined directly from the potential energy function MD has intrinsic dynamics(time dependence) MC dynamics has to be prescribed there is no time scale 7

8 Theoretical and Computational Biophysics Group NIH Resource for Macromolecular Modeling and Bioinformatics Beckman Institute, University of Illinois at Urbana-Champaign VMD molecular visualization programfor displaying, animating, and analyzing large biomolecular systems using 3-D graphics and built-in scripting NAMD parallel molecular dynamics codedesigned for high-performance simulation of large biomolecularsystems. Based on Charm++ parallel objects, NAMD scales to hundreds of processors on high-end parallel platforms and tens of processors on commodity clusters using gigabit ethernet. RCSB Protein Data Bank Information portal to biological macromolecular structures PDB standardized format for macromolecular structure description 8

9 Average number of ion passages as a function of pore radius Voltage-current curves for CNT (10,10) Net ion currents for NaIexceed by up to 30% those for NaCl anion selectivity. Partial Na + currents are anion-independent; polarizability is not relevant. 9

10 C 36 C 60 E bind = 9.75 ev E bind = ev C 70 C 96 E bind = ev E bind = ev 10

11 100% fragmentation 0% fragmentation Fragmentation probability(for each parameter combination) ratio of dissociative trajectories to total number of trajectories Phase transition: fragmentationless regime (p = 0) saturation regime (p = 1) 11

12 Fit of critical points: E = an bq = (3 / 2) an bq 2 2 crit bond tot atom tot N bond = (3 / 2) N atom a = 1 ev, b = 0.33 evfor C 36 b = 0.2 evfor C 60, C 70, C 96 Critical charge fragmentation probability 0.5 without excitation: q crit = ( a / b) N bond Average critical excitation energy summation over integer charges: E = / 2 crit an atom bqtot E crit depends linealy on the fullerene size! E crit = 57.9 ev (55 ev by numerical averaging along the whole critical line) E crit = 34.0 ev (C 36 ), 67.8 ev (C 70 ), 93.4 ev (C90) 12